# About inertial reference frames and logical deduction

• I
• cianfa72
In summary, according to the logic presented, the physical bodies in the inertial frame are not required to have zero coordinate acceleration.f

#### cianfa72

TL;DR Summary
logical deduction about inertial reference frames relative velocities
Hi,

consider the following in the context of classic mechanics and SR.

We know there exist special "frame of reference" according to free objects stay at rest or keep moving with constant uniform velocities. Suppose you single out a such reference frame according to the Newton law of inertia holds for a free object.

Now, from a purely logical point of view (just using logic reasoning), we cannot conclude that the rest reference frame of that free object is actually an inertial one. We need to add a further rule, namely the Galileo principle of relativity, to conclude that.

Does it make sense ?

Summary:: logical deduction about inertial reference frames relative velocities

Suppose you single out a such reference frame according to the Newton law of inertia holds for a free object.
I think you need at least three free objects to define an inertial frame this way.

We need to add a further rule, namely the Galileo principle of relativity, to conclude that.
You could use Galilean relativity for that if you want.

It is not necessary. You could instead simply check to see if the law of inertia works in the new frame also.

Last edited:
• FactChecker and etotheipi
Suppose you single out a such reference frame according to the Newton law of inertia holds for a free object.

How would you distinguish the one you've chosen from all the others? They're equivalent. That's the point of the Principle of Relativity.

How would you distinguish the one you've chosen from all the others? They're equivalent. That's the point of the Principle of Relativity.

There's no preferred frame, but surely you can pick one, pick another one moving inertially w.r.t. this one, and note that the velocities of the same particle measured in both frames differ by a constant vector. That's enough to know that the two frames are different (even though physics doesn't care which one you pick).

• vanhees71
If Newton's laws work in one frame, in order to prove they work in another frame, logically one must be able to transform results from one frame to another.

In the Newtonian case, one uses the Gallilean transform, ##x' = x - vt \quad t' = t##. With this assumption about how coordinates transform, the Newtonian results follow.

In the SR case, one uses the Lorentz transfor, ##x' = \gamma(x - vt) \quad \t' = \gamma(t - vx/c^2)##. Again, with this assumption about how objects coordinats transform, one can prove that SR work in any frame. Alternatively, one might assume that the laws of SR are universal, and then find the restrictions on the coordinate transforms that are possible. This later approach is more common, I think. In this case, one isn't proving the result so much as assuming it, and placing limits as to what transfoormation laws are possible.

I think you need at least three free objects to define an inertial frame this way.
I believe you have in mind Tait's construction to build up it

How would you distinguish the one you've chosen from all the others? They're equivalent. That's the point of the Principle of Relativity.
If I take it correctly, basically you're saying that as I single out one inertial frame I can single out all of them as well

If Newton's laws work in one frame, in order to prove they work in another frame, logically one must be able to transform results from one frame to another.

In the Newtonian case, one uses the Gallilean transform, . With this assumption about how coordinates transform, the Newtonian results follow.

That's my point: starting from the "initial" inertial reference frame Galileo principle of relativity make sure that the rest frame of the free body is actually an inertial one. As far as I can tell Galileo principle of relativity just imposes a linear transformation between coordinates of the two inertial reference frames

Last edited:
As far as I can tell Galileo principle of relativity just imposes a linear transformation between coordinates of the two inertial reference frames
I think conceptually it's better to distinguish between:

a) Galilean relativity (laws are the same in all inertial frames)
b) Galilean transformation (tells you how to transform between frames)

Special relativity still retains a), but replaces b) with the Lorentz-Transformation.

I think conceptually it's better to distinguish between:

a) Galilean relativity (laws are the same in all inertial frames)
b) Galilean transformation (tells you how to transform between frames)

Special relativity still retains a), but replaces b) with the Lorentz-Transformation.
Sure, I believe that's a fundamental point.

Another question related to the first one.
Starting let me say from the definition of inertial reference frame as:" free objects stay at rest or keep moving with constant uniform velocities respect to it" I was wondering why, from a logical argument, the physical bodies building the inertial reference frame are themselves unaccelerated (zero proper acceleration as measured by accelerometers at rest with each of them) when the definition requires just the zero coordinate acceleration of the free bodies as measured in it.

A point to be highlighted is that, in the definition above, I'm assuming we're able to measure forces indipendently avoiding circular arguments in it

Last edited:
Whether or not a reference frame is inertial or not you can of course only establish by observations, i.e. by measuring whether a free body moves always with constant velocity relative to the chosen reference frame.

In practice it's also a question to which accuracy you measure. E.g. usually we work in reference frames at rest relative to Earth, and very often we can neglect that it's not an inertial frame. Looking a bit more accurately, of course you can recognize this, as demonstrated by the often demonstrated Foucault pendulum experiment.

• etotheipi
Looking a bit more accurately, of course you can recognize this, as demonstrated by the often demonstrated Foucault pendulum experiment.

Actually, it's much easier to recognize that a reference frame at rest relative to Earth is not inertial: just drop a rock. What the Foucault pendulum shows is that not only is such a frame non-inertial in the sense of being linearly accelerated (a free body accelerates downward relative to the frame) but also in the sense of being rotating.

• vanhees71 and etotheipi
True, but has anybody really demonstrated this in a real free-fall experiment? Foucault pendulae are quite often found somewhere at most physics buildings ;-).

• etotheipi
has anybody really demonstrated this in a real free-fall experiment?

Demonstrated what? That a dropped rock accelerates downwards? I did that one in my high school physics class. Not with very accurate measurements, but enough to show coordinate acceleration.

• etotheipi
Demonstrated what? That a dropped rock accelerates downwards? I did that one in my high school physics class. Not with very accurate measurements, but enough to show coordinate acceleration.

In GR that would be sufficient, since the free-falling body has non-zero coordinate acceleration but zero proper acceleration.

I don't know if this was what @vanhees71 was referring to, but with the with the classical definition of an inertial frame, in order to show the Earth's surface constitutes a rotating frame we would need to, for instance, measure a deflection of the path of the falling object due to the Coriolis force. That particular experiment seems quite hard to realize, but using a Foucault pendulum demonstrates the effect in question very simply.

Last edited by a moderator:
Aren't all three of Newtons laws needed so inertial coordinate systems are homogeneous and isotropic with respect to inertia?

A point to be highlighted is that, in the definition above, I'm assuming we're able to measure forces indipendently avoiding circular arguments in it

I've never really believed that "circular" arguments can be avoided in physics. Physical theories, like Newton's three laws, tend to come in a package. The three laws essentially introduce several physical quantities: time, space, position, velocity, momentum, acceleration, mass and force and the relationship between them. This whole package gives you a theory that (hopefully) is experimentally testable. Moreover, I would say that Newton's laws are part of theoretical physics.

Actually testing the theory is a different matter. That's experimental physics. In principle, there is no limit to how clever you might have to be to test a theory. Testing Newton's laws may not be too bad, but you have to make several practical decisions about what sort of set-up constitutes a valid test. It's also not that easy to justify when exactly you have a valid test. An Aristotelian might be able to provide a lot of experimental evidence in support of an alternative theory. And a lot of evidence against Newton's laws.

I don't believe there can be a mathematical-like perfection in theoretical and experimental physics. Establishing an inertial reference frame and testing Newton's first law, for example, may require a lot of human ingenuity that is not at all specified by the theory.

• Dale
I don't believe there can be a mathematical-like perfection in theoretical and experimental physics. Establishing an inertial reference frame and testing Newton's first law, for example, may require a lot of human ingenuity that is not at all specified by the theory.
Sure, I was just trying to address it form a theoretical/axiomatic point of view if any.

My attempt is the following:
1. take a *free* body and define its state of motion as inertial: the special property of this state of motion is the following: an accelerometer attached to it measure zero value (zero proper acceleration)
2. define inertial frame of reference as a frame in which any *free* body shows zero *coordinate* acceleration as measured in it
3. take one of that free bodies in (2) and consider the reference frame at rest with it: thanks to Galileo principle of relativity we can conclude that it is inertial as the first one
4. from the point of view of the inertial reference frame in (3) points (or bodies) at rest in the first one move with constant uniform velocity (zero *coordinate* acceleration) and in force of Newton law of inertia - basically the content of (2) - we can conclude they are *free* and in turn accelerometers attached to each of them measure zero (proper) acceleration

Sure, I was just trying to address it form a theoretical/axiomatic point of view if any.

My attempt is the following:
1. take a *free* body and define its state of motion as inertial: the special property of this state of motion is the following: an accelerometer attached to it measure zero value (zero proper acceleration)
2. define inertial frame of reference as a frame in which any *free* body shows zero *coordinate* acceleration as measured in it
3. take one of that free bodies in (2) and consider the reference frame at rest with it: thanks to Galileo principle of relativity we can conclude that it is inertial as the first one
4. from the point of view of the inertial reference frame in (3) points (or bodies) at rest in the first one move with constant uniform velocity (zero *coordinate* acceleration) and in force of Newton law of inertia - basically the content of (2) - we can conclude they are *free* and in turn accelerometers attached to each of them measure zero (proper) acceleration
Is this a theoretical or experimental construction?

I've never really believed that "circular" arguments can be avoided in physics.
I would say there is a difference between "arguments" and "definitions".

Physical theories, like Newton's three laws, tend to come in a package.
Exactly, you postulate and define several things together. You don't have to derive one postulate from another.

Is this a theoretical or experimental construction?
I think theoretical in the first part; then, in order to check if conditions apply, you have to do experiments - for instance to check if a a body is *free* you have to attach to it an accelerometer and do a measure

• Dale
I think theoretical in the first part; then, in order to check if conditions apply, you have to do experiments - for instance to check if a a body is *free* you have to attach to it an accelerometer and do a measure
Would you say that is how experimental physics is done? How do you attach an accelerometer to a particle?

For example, if you were in the ISS and wanted to do an experiment would you a) do a series of preparatory tests and experiments to check that is in indeed inertial frame; or, b) assume provisionally that it is inertial and confirm that the results of experiments agree with what would be expected in an inertial frame?

Would you say that is how experimental physics is done? How do you attach an accelerometer to a particle?

For example, if you were in the ISS and wanted to do an experiment would you a) do a series of preparatory tests and experiments to check that is in indeed inertial frame; or, b) assume provisionally that it is inertial and confirm that the results of experiments agree with what would be expected in an inertial frame?
Not sure to understand your point (sorry I'm not a physicist). Suppose also you are in the ISS: why can't you *in principle* attach an accelerometer to a body to check if it is free (zero value for the accelerometer measure) ?

Not sure to understand your point (sorry I'm not a physicist). Suppose also you are in the ISS: why can't you *in principle* attach an accelerometer to a body to check if it is free (zero value for the accelerometer measure) ?
I think my point was that, in practice, a well-designed experiment would confirm both the nature of the reference frame and Newton's laws as an inference from the results. Rather than follow an elaborate process to establish these separately. Again, it's like a package: you suspect you have an IRF and Newton's laws and the test confirms this (as far as it is able).

A good example, perhaps, is the time-dilation on GPS satellites. Effectively you have a system test: a test of everything at once: all the assumptions about SR and GR and everything else go into the equations; a test is done and total differential time is measured and compared with the theoretical total. And, you try to do multiple test with varying parameters. I can't see any way you could directly and categorically establish that we have the Schwarzschild geometry around the Earth. You could in principle imagine all sorts of sophisticated tests to establish that. Instead, that assumption is thrown into the theoretical mix and the whole package of theory is tested.

I think my point was that, in practice, a well-designed experiment would confirm both the nature of the reference frame and Newton's laws as an inference from the results. Rather than follow an elaborate process to establish these separately. Again, it's like a package: you suspect you have an IRF and Newton's laws and the test confirms this (as far as it is able).
Based on what you said, I think my formulation in post #16 is a theoretical/axiomatic one in which the *free* body condition is established through an *operative definition* - namely zero reading of accelerometers at rest with bodies

I think you need at least three free objects to define an inertial frame this way.
Could you elaborate a bit, please ? If we take just two bodies alone can we always find out a reference frame (a system of objects at rest each other) in which those two bodies undergo zero *coordinate* acceleration ?

Last edited:
Demonstrated what? That a dropped rock accelerates downwards? I did that one in my high school physics class. Not with very accurate measurements, but enough to show coordinate acceleration.
I talked about the deviation to the east (and far smaller south) due to the Coriolis force due to rotation of the Earth around its axis.

From the point of view of GR it's of coarse already sufficient to have the (nearly) constant gravitational acceleration ##g=9.81 \text{m}/\text{s}^2## to see that we are not in a (local) inertial frame, and it's sufficient to observe that bodies "fall down" on Earth.

Coming back to definition of inertial reference frame, I found that up to three bodies we are always able to single out a *rigid* reference frame (basically three orthogonal axis starting from a common origin) for which bodies follows straight paths

Thus following the idea of Lange, we can assume (in the context of Newtonian mechanics and SR) an inertial reference frame is actually defined by three *free* bodies and that law of inertia actually amounts to establish the behavior of the fourth and subsequent *free* bodies (namely that in the reference frame just built their paths are actually straights)

• vanhees71
up to three bodies we are always able to single out a *rigid* reference frame (basically three orthogonal axis starting from a common origin) for which bodies follows straight paths

Can you give a reference?

Can you give a reference?
You can see the work of D. Giulini at Traegheit --- it's in german but you can just focus on the last two-page at section 10.

It should be a review of Lange's original work namely On the law of Inertia -- see it here

• Dale and vanhees71